$\DeclareMathOperator{\Spec}{Spec}$ $\DeclareMathOperator{\im}{im}$ Hello Math.Stackexchange.com-Community.
Sorry for asking two questions at the same time, however they are part of one single step in a proof and hence closely related. My questions arise from the following context:
Let $X\to Y$ be a proper morphism between locally noetherian Schemes and let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module on $X$ which is flat over $\mathcal{O}_Y$. Then for all $p\geq 0$ the function $h^p(-,\mathcal{F})\colon Y\to\mathbb{N}$ which is defined by $h^p(y,\mathcal{F}):=\dim_{\kappa(y)}H^p(X_y,\mathcal{F}|_{X_y})$ is upper semi-continuous in the sense that for all $c\in\mathbb{R}$ the set $\{y\in Y\mid h^p(y,\mathcal{F})\leq c\}$ is open.
In order to prove this, the following lemma was invoked:
Lemma: Let $A$ be a noetherian ring. Let $\delta\colon M\to F$ be a morphism between finitely-generated $A$-modules, of which $F$ is free. Then, the function $g\colon \Spec(A)\to \mathbb{N}$ defined by $g(\mathfrak{p}):=\dim_{\kappa(\mathfrak{p})}(\ker(M\otimes_A \kappa(\mathfrak{p})\overset{\delta_{\mathfrak{p}}:=\delta\otimes_A \kappa(\mathfrak{p})}{\to} F\otimes_A \kappa(\mathfrak{p}))$ is upper semi-continuous.
My question arises in the proof of this lemma, which goes as follows:
Proof Let $\mu_1,\cdots,\mu_s\in M$ be elements, such that the images $\overline{\mu_1},\cdots,\overline{\mu_s}\in M\otimes_A\kappa(\mathfrak{p})$ form a basis of that vector space, and such that $\delta_{\mathfrak{p}}(\overline{\mu_i})$ vanish for $i=1,\cdots,r=g(\mathfrak{p})$ and form a basis of $\im(\delta_{\mathfrak{p}})$ for $i=r+1,\cdots,s$.
Here comes the part that I don't understand, and also my first question:
Since the assertion is local near any $\mathfrak{p}$ we may replace $A$ by $A_f$ for $f\not \in\mathfrak{p}$ and assume that $M$ is generated by the $\mu_1,\cdots\mu_s$ as an $A$ module.
The last assertion is clear to me. This is a Nakayama-Style argument. It is however not clear to me, what "local near $\mathfrak{p}$" means, and why this implies, that we may replace $A$ by $A_f$. Aside from this argument, it is also not clear to me, how to prove by hand (i.e. without using the "near $\mathfrak{p}$"-Argument) why we may replace $A$ by $A_f$.
I tried and failed as follows:
Assume that we may find a covering by $D(f_j)$'s of $\Spec(A)$ such that the morphisms $g_j\colon Spec(A_{f_j})\to\mathbb{N}, g_j(\mathfrak{p}_j)=\dim_{\kappa(\mathfrak{p}_j)}(\ker(M_{f_j}\otimes_{A_{f_j}}\kappa(\mathfrak{p}_{f_j})\to F_{f_j}\otimes_{A_{f_j}}\kappa(\mathfrak{p}_{f_j}))$ is EDIT: locally constant upper semi-continuous, where by $\mathfrak{p}_j$ I mean the prime ideal in $\Spec(A_{f_j})$ corresponding to $\mathfrak{p}\in D(f_j)$.
Then it would suffice if one could show that this implies, that the restrictions of $g$ to the $D(f_j)$ are upper semi-continuous.
In order to show this one would have to show, that for every $c\in\mathbb{R}$ we have $\dim_{\kappa(\mathfrak{p}_j)}(\ker(M_{f_j}\otimes_{A_{f_j}}\kappa(\mathfrak{p}_{f_j})\to F_{f_j}\otimes_{A_{f_j}}\kappa(\mathfrak{p}_{f_j}))\leq c$ for all $j$ iff $\dim_{\kappa(\mathfrak{p})}(\ker(M\otimes_A \kappa(\mathfrak{p})\overset{\delta\otimes_A \kappa(\mathfrak{p})}{\to} F\otimes_A \kappa(\mathfrak{p}))\leq c$ for all prime ideals $\mathfrak{p}\in D(f_j)$.
This is the point where I don't see how to continue.
My second question is why we may construct a basis for $M_{f_j}\otimes_{A_{f_j}}\kappa(\mathfrak{p_{f_j}})$ out of the basis for $M\otimes_A \kappa(\mathfrak{p})$ such that it still has the properties we set up in the beginning of the proof, so that we may carry on with the proof after replacing $A$ by $A_f$.
Thank You for your participation!
Let me answer your first question. I will use the following setup: $A$ a noetherian ring, $M$ a finitely generated $A$-module, and $p$ a prime ideal of $A$. We show that if $M_p$ is generated by the images of $x_1,\dots, x_n \in M$, then there exists $f \in A$ such that $M_f$ is generated by the images of the same set of elements.
Let $N$ be the $A$-submodule of $M$ generated by $x_1,\dots, x_n$, and let $K$ be the cokernel of the inclusion map $N \to M$. Our goal is to find an element $f \in A$ such that $K_f = 0$. If $K$ is $0$, then there is nothing to do. Assume that $K \neq 0$. Consider the support of $K$. Since $K$ is finitely generated $Supp(K)$ is determined by an $A$-ideal $I$, and since $p$ is not in it, $I$ is a proper ideal. Hence we may choose any nonzero element in $I$ for $f$.